Question

1. why does hl (hypotenuse – leg) work as a triangle congruence criterion?

Explanation:

HL applies to right - angled triangles. In a right - angled triangle, by the Pythagorean theorem ($a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse and $a,b$ are the legs), if the hypotenuse and one leg of two right - angled triangles are equal, we can calculate the other leg. Let the hypotenuse of triangle 1 be $c_1$, leg $a_1$; hypotenuse of triangle 2 be $c_2$, leg $a_2$. Given $c_1 = c_2$ and $a_1=a_2$, then for triangle 1, $b_1=\sqrt{c_1^{2}-a_1^{2}}$, for triangle 2, $b_2=\sqrt{c_2^{2}-a_2^{2}}$. Since $c_1 = c_2$ and $a_1=a_2$, $b_1 = b_2$. Now we have all three sides equal (SSS) or two sides and the included right angle equal (SAS), so the triangles are congruent. So HL works because the Pythagorean theorem ensures the third side is equal, leading to SSS or SAS congruence for right - angled triangles.

Answer:

HL (Hypotenuse - Leg) works for right - angled triangles. Using the Pythagorean theorem ($a^{2}+b^{2}=c^{2}$), if hypotenuse ($c$) and one leg ($a$) of two right - angled triangles are equal, the other leg ($b=\sqrt{c^{2}-a^{2}}$) must also be equal. This gives SSS (all three sides equal) or SAS (two sides and included right angle equal) congruence, so the triangles are congruent.